3 Answers
3

The idea was to raise Theon as a member of the Stark family, so that he would come to understand and respect the ways of Winterfell. That included allowing him to ride around the woods and handle weapons. This was common practice with ancient and medieval hostage taking. From http://en.wikipedia.org/wiki/Hostage :

The practice of taking hostages is very ancient, and has been used
constantly in negotiations with conquered nations, and in cases such
as surrenders, armistices and the like, where the two belligerents
depended for its proper carrying out on each other's good faith. The
Romans were accustomed to take the sons of tributary princes and
educate them at Rome, thus holding a security for the continued
loyalty of the conquered nation and also instilling a possible future
ruler with ideas of Roman civilization.

Theon could have tried to kill Eddard or one of the Stark children to avenge his brothers. But if he did so, he would probably be hunted down by the surviving Starks or their bannermen. It's a very long way from Winterfell to the Iron Islands; Theon's chances of making the journey as a fugitive who was wanted for murdering a Stark would be rather slim.

Not only that, he would have dishonoured his father and likely provoked a war between the Iron Islands and the rest of the Seven Kingdoms -- a war which, prior to the death of Robert Baratheon, the Iron Islands would almost certainly lose. So he would have been less than welcome if he reached the Iron Islands. Theon knew all this, and in any event he seems to have liked and respected Eddard and his children, having been taken to Winterfell at a very young age.

Finally, there is no indication in the books that Eddard fully trusted Theon. Eddard was never alone with Theon, and he didn't send Theon far away from Winterfell, much less order him to visit the Iron Islands. As Thundergr points out, that was Robb's mistake.

"In any event he seems to have liked and respected Eddard and his children, having been taken to Winterfell at a very young age." I got the impression from the books that he rather enjoyed the Starks, until his father disowned him.
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Mooing DuckFeb 13 '14 at 20:13

I didn't know people actually used to do that! That's really interesting
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ShevliaskovicFeb 13 '14 at 21:08

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@MooingDuck: Yes, indeed. But during all his years with the Starks, Theon dreamed of a hero's welcome when he returned to the Iron Islands. When he eventually turned up there, he had a very rude awakening. At that point he didn't have any good options, but the course of action he settled on was particularly destructive (to Theon himself, and everyone else involved).
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Royal Canadian BanditFeb 13 '14 at 23:02

The boy was taken very young. He was raised more like a member of the family rather than a hostage. Robb and Theon were very close. In the end, it was not Ned that trusted him, though. It was Robb's poor judgement...

It is interesting that Theon did not want to harm the Starks, until he got close to his father and sister again and felt the need to prove himself to them and their plans.

Yes. When Theon reaches the Iron Islands, not only is he treated with utter contempt by his father, his sister, and pretty much everyone else; he also learns that Balon plans to make war on Winterfell regardless. At that point, going back to Robb and announcing his mission has failed is not an option. A more prudent man might have tried to obey Balon, ride out the war as best he could, and hope to gain acceptance among the Ironborn over time, but that's not Theon.
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Royal Canadian BanditFeb 13 '14 at 17:16

@RoyalCanadianBandit A really honorable person, worthy of the trust Robb had showed to him and the upbringing Ned gave him, would make haste to Winterfell and warn about Balon's plans. Gaining the respect, trust and loyalty of the North. But, again, that's not Theon :)
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ThunderGrFeb 13 '14 at 17:39

Balon would certainly not have let Theon go willingly. If Theon had somehow gotten back to Robb, Balon would probably have gone to war anyway. At that point even Robb would not have been so foolish as to trust Theon to remain loyal to him rather than his blood family. At best, Theon would be imprisoned by Robb for the foreseeable future; and he would have permanently lost any chance of trust or respect from the Ironborn. All in all he would be facing a pretty bleak future. What you suggest might be honourable but it would also be rather stupid.
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Royal Canadian BanditFeb 13 '14 at 17:45

@RoyalCanadianBandit: Given his ultimate fate, a bleak future would probably have been the better choice. But, as you alluded to, Theon is not known for his wise decisions.
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OmegacronFeb 13 '14 at 19:33

@RoyalCanadianBandit I think you underestimate the Starks. If Theon had gone to Winterfell and warned them, they would had accepted him with open arms. Balon would go to war, sure. But Theon would had proven himself and gained the full trust of the Starks, for certain. When reading the book, I was hoping he'd do that. Theon had a lot of potential, if only he could break free from his ambition.
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ThunderGrFeb 13 '14 at 19:34

Ned could trust him as long as Theon was at Winterfell, since everyone there is very loyal to the Starks and would still remember the Greyjoy rebellion. Theon could not do much to undermine the Starks while in that position, even if he wanted to.

Notice that Ned did not take Theon south with him when he went to King's Landing. He probably did not trust him that much.

Finally, for what it's worth, not everyone else trusted Theon. Sure, he was good friends with Robb, but Jon Snow never liked Theon very much. Although that could have been due in large part to both of them being outsiders of sorts at Winterfell.